Respuesta :
Answer:
Steve should use:
- 8.53 Inch of Wire to make the circle
- 43.47 Inches of Wire to make the Square.
Step-by-step explanation:
Let Steve cut the wire so that the first piece has length x.
Therefore, the second piece will have a length of (52 - x).
The wire of length x is used to make a circle
Circumference of a Circle,
[tex]c = 2\pi r\\Therefore\\2\pi r=x\\r=\dfrac{x}{2\pi}[/tex]
[tex]\text{Area of a circle, A}=\pi r^2= \pi (\dfrac{x}{2\pi})^2=\pi (\dfrac{x^2}{4\pi^2})\\A=\dfrac{x^2}{4\pi}[/tex]
The wire of length (52-x) is used to make a square.
[tex]\text{Side length of the Square,}[/tex] [tex]s= \dfrac{52-x}{4}[/tex]
[tex]\text{Area of the Square},s^2= \left(\dfrac{52-x}{4}\right)^2=\dfrac{(52-x)^2}{16}=\dfrac{x^2-104x+2704}{16}[/tex]
Total Area = Area of Circle + Area of Square
[tex]Area=\dfrac{x^2}{4\pi}+\dfrac{x^2-104x+2704}{16}[/tex]
Let us simplify the expression
[tex]Area=\dfrac{16x^2+\pi(x^2-104x+2704)}{16\pi}\\=\dfrac{16x^2+\pi x^2-104\pix+2704\pi}{16\pi}\\=\dfrac{x^2(16+\pi)-104\pi x+2704\pi}{16\pi}\\=\dfrac{x^2(16+\pi)}{16\pi}-\dfrac{104\pi x}{16\pi}+\dfrac{2704\pi}{16\pi}\\A=\dfrac{x^2(16+\pi)}{16\pi}-6.5x+169[/tex]
This is the function of a parabola which opens up.
To find where A is minimum, find the axis of symmetry.
[tex]$Using \: x=-\frac{b}{2a}[/tex]
[tex]a=\dfrac{16+\pi}{16\pi}, b=-6.5\\ x=-\dfrac{-6.5}{2(\dfrac{16+\pi}{16\pi})}=8.53\:Inches[/tex]
Steve should cut the wire so that the length of wire used to make a circle is 8.53 Inches.
Length of wire used to make the square =52-8.53=43.47 Inches
